A note on Fibonacci and Lucas number of domination in path

Abstract

Let G = (V(G), E(G)) be a path of order n ≥ 1. Let fm(G) be a path with m ≥ 0 independent dominating vertices which follows a Fibonacci string of binary numbers where 1 is the dominating vertex. A set F(G) contains all possible fm(G), m ≥ 0, having the cardinality of the Fibonacci number Fn + 2. Let Fd(G) be a set of fm(G) where m = i(G) and Fdmax(G) be a set of paths with maximum independent dominating vertices. Let lm(G) be a path with m ≥ 0 independent dominating vertices which follows a Lucas string of binary numbers where 1 is the dominating vertex. A set L(G) contains all possible lm(G), m ≥ 0, having the cardinality of the Lucas number Ln. Let Ld(G) be a set of lm(G) where m = i(G) and Ldmax(G) be a set of paths with maximum independent dominating vertices. This paper determines the number of possible elements in the sets Fd(G), Ld(G), Fdmax(G) and Ldmax(G) by constructing a combinatorial formula. Furthermore, we examine some properties of F(G) and L(G) and give some important results.

Keywords

• fibonacci numbers, lucas numbers, fibonacci path domination, lucas path domination, independent domination